Resonant state selection in synthetic ferrimagnets B. C. Koop,1 Yu. I. Dzhezherya,2 K. Demishev,2 V. Yurchuk,2 D. C. Worledge,3 and V. Korenivski1, ∗ 1 Royal Institute of Technology, 10691 Stockholm, Sweden Institute of Magnetism, Ukrainian Academy of Sciences, Kiev, Ukraine 3 IBM T. J. Watson Research Center, Yorktown Heights, NY 10598, USA (Dated: February 28, 2013)
arXiv:1302.6483v2 [cond-mat.mes-hall] 27 Feb 2013
2
Resonant activation of a synthetic antiferromagnet (SAF) is known to result in a dynamic running state, where the SAF’s symmetric spin-flop pair continuously rotates between the two antiparallel ground states of the system, with the two magnetic moments in-phase in the so-called acoustical spinresonance mode. The symmetry of an ideal SAF does not allow, however, to deterministically select a particular ground state using a resonant excitation. In this work, we study asymmetric SAF’s, or synthetic ferrimagnets (SFi), in which the two magnetic particles are different in thickness or are biased asymmetrically with an external field. We show how the magnetic phase space of the system can be reversibly tuned, post-fabrication, between the antiferro- and ferri-magnetic behavior by exploiting these two asymmetry parameters and applying a uniform external field. We observe a splitting of the optical spin-resonance for the two ground states of the SFi system, with a frequency spacing that can be controlled by a quasistatic uniform external field. We demonstrate how the tunable magnetic asymmetry in SFi allows to deterministically select a particular ground state using the splitting of the optical spin-resonance. These results offer a new way of controlling the magnetic state of a spin-flop bilayer, currently used in such large scale applications as magnetic memory. PACS numbers:
I.
INTRODUCTION
Magnetic memory technology employs the effects giant magnetoresistance1,2 and tunneling magnetoresistance3,4 in spin-valves type sensors5 . In these, the switching of the magnetic state is achieved by using either external magnetic field6–9 or spin-transfer-torque10–13 writing. In this work we demonstrate a reliable way to switch a magnetic nanodevice employing microwave excitation in resonance with the spin eigen-modes in the device, which act to amplify the action of the microwave field. Synthetic antiferromagnets consist of two identical dipole-coupled magnetic layers separated by a thin spacer. The dipole coupling results in an antiparallel (AP) alignment of the magnetic moments in the ground state, which makes them similar to classical atomic antiferromagnets but with a tunable coupling strength between the magnetic moments set in fabrication by a suitable choice of the material parameters and geometry. When the lateral shape is elliptical, the system has two AP ground states, in which the moments are aligned along the easy-axis (EA) of the ellipse. Due to the symmetry, the energy of the two AP ground states is degenerate. The quasistatic1,14,15 and dynamic17–20 behavior of SAFs has been studied extensively. Due to the nearly perfect flux closure in the AP state, SAFs are characterized by high stability against thermal agitation21 . It was recently shown that the spin dynamics contains collective acoustical and optical resonance modes22 . It has also been shown that excitation at the optical resonance frequency can result in switching between the two AP ground states23 , typically resulting in multiple switching
or a spin-running state, since the resonant switching in an ideal SAF is symmetric for the two AP ground states so a specific state cannot be selected deterministically. In order to enable use of resonant excitation for selecting a specific ground state, which is highly desirable for applications, the spin-resonance spectrum can be tuned such that, for example, resonant switching occurs at different frequencies for the two ground states. For this the spin-flop bilayer must have suitable magnetic asymmetry, such as asymmetry in the the magnitude of the two magnetic moments or asymmetric field biasing of the layers externally so each layer experiences a different Zeeman field. The former can be achieved by making the two layers of the same material but slightly different thickness or of the same thickness but of materials with different saturation magnetization. The latter can be built-in into the nominally flux-closed reference layer of the read out junction24 . These two magnetic asymmetry contributions can also be combined, which we do in this study for achieving greater flexibility in controlling the energetics and spin dynamics in the SFi system. In this paper we develop a method of deducing the individual magnetic asymmetry contributions in SFi, postfabrication, and show how they can be used to reversibly tune the magnetic behavior of the system between SAFlike and SFi-like. We further show how an external uniform magnetic field can be used to control the symmetry of the magnetic double-well potential as well as the splitting of the optical spin resonance in SFi with a thickness imbalance and asymmetric local field-bias.
Hx =
t1 = t2
tm2 H1m t6= 1 H2 m t H 2 2 Hx =t1 = mt1
(14) H1 6= H2 (13) t2 H m t2 H2mHx = t1 H1m 2 Hx = t1 (14) t1 t2 (15)
t2t1 H1m t2
H1 6= H2m
(13)
t1 H1m
(14) (15)
t2
(14)
t1 6= t2
(4)
H1m = H2m
(16)
(5)
(1)
⇡)
E ('1 = '2 Amplitude
1
1
'1 = 0 '1 = ⇡
!1
!2
!1 = !2
Amplitude
'1 = 0 '1 = ⇡
'1 = 0 '1 = ⇡
!1
!2
!1 = !2
= '2 ⇡) E!(' Amplitude !⇡ !1'1!=2 0 !' !1 !2 !1 = 2 1= 2 1 Amplitude 1=
degenerate resonance resonance splitting
Resonance behavior
E(1)('1 = '2
⇡)
E ('1 = '2
⇡)
(1)
(1)
2 Hx = 0 (6) H1m 6= H2m (16) t1 = t2 (15) t1 m = t (17)m )m (15) g)s (t1 , t2 , H (17) m Hx 6= tg1s (t , Hx1m ,6= H2m =1t, 2t2H 1m , 2H 2 (15) (16) m H1 6= H2 H1 6= (16) mH2 H(16)6= g (t , t , H m , H m ) t2m Hm H 6= H2tm1 H1m t H m (17) tH x ms 1m 2 2 2 1 H6=1 g(18) 1 2 Hx 6= 1 2 Potential energyH = g (t , t , H m , H m ) ) (17) x s (t1 , t2 , H1 , Hm (7) x s 1 2 m t1 gH t21x, t26= 1 2 (18) Hx 6= , H1m , H2m ) (17) t2 H 2 s (t t1 H1 m m 2 Hardt1 H1 (18) t2 H2m potential t1 H1m tH Symmetric potential Asymmetric 2 x 6= t2 H2Hx 6= ' (t) (1) t2 H2m t1 H1m t1 1 axis y (18) t t (8) H = 6 1 2 x m (18)t2 t1 =t1t2 t2 (1) H1m HxH6= top free layer M2(19) t1 2 tt11 t2t2 M1 (1)t1 = t2 (1) (1) M1 m m ' (t) (2) H = H (2) m m 2 t2 1 2 m m H H t (19) m m 1 TaN spacer m Easy1 2 (2) H Ht2m t2 2 (19) 2 H t11 M (2)H1 = H(2) 2(19) t1 (1) M2'0 1 (t)(19) |Hx | < Hs (3) 1 H1m H2m t1 bH axis ottt2 2 0 om free layer M1 t1(3) 6= t2 (9) |Hx | < Hs '2 (t) x '1 (t) m m = (3) AlO tunnel barrier H = 6 H (10) x 0! 0Amplitude ' (t) (2) 1 2 2 ' = 0 ' = ⇡ ! ! = ! E (' = ' ⇡) 2⇡ (1) |' (t)| |' (t)| 1 1 1 1 2 1 2 2 1 2 top fixed layer M 0 t2 H m t1 H1m '1 = 0 '1 = ⇡ !1 !2 !1 = !2 Amplitude0 E ('1 = '2 0 ⇡) 2⇡ (1) t1 = t2 (4) Hx =t1 6= t22 (1) (11) 0 '1 (t)'0 (t) WordBit-'2 (t) t1 6= t2 (4) t1 mt2 coupling layer (t) ' m m 1 1 Energy as a function 2 H1 = (3) H2 (2) =line 0 of in-plane H1m = H (5) line (2)m 2 bottom fixed layer m = (4) 0 H1 = H2 (5) 0 |' (t)| |' 2 (t)| |H 0)
11
1
0.5 0.5
M1x / MS
splitting in the optical eigen-mode of the system and is given by γf = hd /Ω2o . The separation of the respective optical spin resonance frequencies is due to the difference in the effective biasing field acting on the individual layers. If the amplitude of the excitation is large enough and the frequency is chosen in resonance with a given ground state, switching into this state occurs, while the reverse switching is offresonance and therefore disallowed energetically. Interestingly, although the oscillation of the two macrospins is of optical type (out-of-phase), the switching occurs acoustically (in-phase rotation). The dipolecoupling is too strong for the moments to cross passed each other under the relatively weak microwave resonant fields used in this work. The switching trajectory is thus an in-phase rotation with superposed out-of-phase oscillations of smaller amplitude, with the energy effectively transferred from the pumped optical oscillations into a large-angle acoustical rotation. As a result, the expected fastest switching time is the half-period of the acoustical resonance frequency (corresponding to approximately 500 ps for our experimental geometry). For still faster resonant switching one would aim in the SFi-device design at maximizing the acoustical resonance frequency while maintaining a suitable size splitting in the optical resonance frequency; for example, by increasing the aspect ratio of the nanomagnets.
20
Start excitation
FIG. 3: (Color online) Stability diagram versus frequencyamplitude for the SFi ground states under resonant excitation. The curves separate the regions of stability and instability (below and above). The greyed out area shows the frequency-amplitude region where only one of the two states is stable, into which the system switches for given excitation. Ωo is the optical center frequency (that of an ideal SAF), hy the amplitude of the excitation, γf = (Ω − Ωc )/Ωc ⇒ γf · 4Ω2o ∝ Ω − Ωc , Ω the excitation frequency.
High→ Low
0 2
Angle with EA
0 0
30
Measured
0.5
0
−0.5
−1 −200
00
−100
ε 0 (main panel), corresponding to our typical experimental configuration (left inset), and for ε < 0 (top-right inset). In contrast, an ideal, symmetric spin-flop bilayer shows a strictly anti-symmetric response (bottom-right inset; simulated).
vious work on the energetics of the SAF system1,28 . The magnetization vectors evolve in time driven by torques of the respective effective magnetic fields, with a suitably small time step of 5 ps. A thickness imbalance and fringing field are the two asymmetry parameters. The simulations include thermal agitation at T = 300 K. In the case of no thickness imbalance, with the only asymmetry due to the unequal biasing fields on the individual free layers (from the reference layer built-in into the nanopillar), the numerically simulated switching behavior agrees well with that predicted analytically above. The optical resonance is clearly split, although the magnitude of the splitting is somewhat 1 1 1 smaller. 1
(1)
(1)
6 A more pronounced resonance splitting is obtained by adding a thickness-imbalance to the magnetic asymmetry of the system. Figure 4a shows a typical simulated switching map. The area where switching occurs only in one direction is where only one ground state is dynamically stable. The general form of the stability regions simulated here for large-signal microwave excitation agrees well with the analytical results. Figure 4b shows the real-time trace of the magnetization of the individual layers during a switching event. It is clear that the microwave-pumped oscillations are of optical nature but the switching itself is an ”acoustical” in-phase rotation. The oscillations of the magnetization in the initial ground state under a resonant excitation of amplitude suitably large to produce switching are, in fact, smaller in the switched-to ground state since this state is now off-resonance with the microwave field. This effect of resonant state selection is a direct consequence of the magnetic asymmetry of SFi. The full macrospin model further yields important insights into the quasistatic magneto-resistance properties of the junctions. Using Eq. S1 the quasistatic switching (H1± ) and spin-flop fields (H2± ) can be obtained and used for extracting the asymmetry parameters (individual layers’ thicknesses and effective biasing fields) from the measured data. The interrelation between the thicknesses and fringing fields is (in notations of Fig. 4(c)): Hi± = ∓(−1)i |ε|(Ny −Nx −γx)−hf 4πMS s 2 ε ± Ny −Nx +γx +hu ±(−1)i hd −(1−ε2 )γy2 , |ε|
|ε| ≈
− + − h+ 1 −h1 −h2 +h2 , 4(Ny − Nx − γx )
hf ≈
− + − −(h+ 1 +h1 +h2 +h2 ) . 4
Here H1± = 4πMS h± 1 is the field for which only one AP ground state becomes unstable and H2± = 4πMS h± 2 is the spin-flop field, at which both AP ground states be± come unstable. Note that the actual values of H1,2 are slightly smaller than the analytical values due to thermal fluctuations. IV.
EXPERIMENTS A.
Samples
The samples used in the experiments were spin-flop type magnetic random access memory (TMRAM) cells with lateral dimensions of 450x375 and 420x350 nm, and elliptical in-plane shape. The stack consists of a magnetically soft SFi separated from a magnetically hard SAF reference layer by a thin Al-O tunnel barrier. The spinflop bilayer is composed of two dipole coupled permalloy
ferromagnets (free layers) with thicknesses t1 ≈ t2 ≈ 5 nm (excluding the magnetically ’dead’ layers at the film surfaces29 ) separated by a ∼1 nm thin TaN spacer, for which there is no interlayer exchange coupling, J = 0. Here t1 and t2 are the effective thicknesses of the bottom and top free layer, respectively. The SFM reference layer is nearly ideally flux-closed and is therefore essentially insensitive to external fields of strengths used in the experiments discussed in this paper. The samples used in this work had a weak fringing field from the reference layer acting on the two free layers. The fringing field originated predominantly from the top fixed layer. The resistance of the stack was ∼1 kΩ, and the magnetoresistance ∼20%.
B.
Measurement setup
In-plane quasi-static fields were applied using an external toroidal magnet30 . High-frequency fields were generated by contacting the on-chip integrated 50 Ω bitand word-lines using surface probes with 0-40 GHz bandwidth. The bit- and word-lines were oriented at ± 45◦ with respect to the the easy-axis of the stack in the socalled ”toggling” configuration. A current through the bit- or word-line results in an in-plane magnetic field perpendicular to the line. In the results presented below, the high frequency excitation was applied using the word line only, which was electrically decoupled from the stack. The applied high-frequency fields were AC pulses of desired frequency ranging from 1 MHz to 10 GHz. The AC fields were generated using an Agilent E8247C PSG CW Signal Generator with a bandwidth of 250 kHz - 20 GHz. The wave-duration was controlled using an RFswitch driven by an Agilent 33250A AWG. The shortest possible pulse-length with this setup was 8 ns. The state of the sample was determined by measuring the resistance of the sample and thereby the relative orientation of the magnetization of the bottom free and top fixed layers. To separate the DC-signal from the RF-signal, a bias-tee was used.
C.
Experimental results
The samples were SFi bilayers of approximately 5 nm thick Permalloy, separated by 1 nm thick TaN, integrated into approximately 350x400 nm in-plane elliptical nanopillars, also containing an Al-O read-out junction with a nominally flux-closed magnetic reference layer. The microwave excitation was applied using a closely spaced field-line making a 45◦ angle with the SFi easy axis. The ferromagnetic resonance frequencies were measured using transport spectroscopy, where the junction resistance versus frequency showed a maximum or a minimum for the low-resistance or high-resistance ground state, respectively. The resistance was measured while slowly sweeping the frequency of a continuous-wave ex-
7
10
20
30
Measured switching probability 300ms excitation, 450x375 nm sample, -20 Oe easy axis field applied
Stability criterion under resonant excitation 0.1 l = ( 2, 0, 0 ) l = ( −2, 0, 0 )
High → Low resistance
h / (4 2p)
Low → High resistance 0
3.6
−
0
4.0 Frequency (GHz) (4
2 p)
4.4
00
5 5
10 10 Time (s) Time (s)
15 15
4.0 3.0 2.0 1.0 0 20 20
d
DC-response of 420x350 nm sample during continuous with of Resistance response of 420x350nm sample duringexcitation a sequence
1
0
f = 4.4 GHz, Hy = 32 Oe, applied EA-field alternates: Hx = -35, 0, 35 Oe
1.9 1.8 1.7 1.6
35 0
-35
32
0 00
20 20
40 40 Time (s) Time (s)
60 60
EA-field (Oe)
60
3.2
1.5
Applied frequency (GHz)
0 EA-field (Oe)
70
50
1.6
Resistance ( )
-10
Resistance (kΩ)
-20
Switch probability
Amplitude (Oe)
80
-30
Amplitude (Oe)
b
1.7
1.4
3.8
3.6
Applied frequency (GHz)
4.0
1.8
Resistance ( )
Resistance (kΩ)
Hy = 58 Oe, f1 = 3.15 GHz, f2 = 3.65 GHz, Hx = -20 Oe
Excitation amplitude
4.2
Experimental optical resonance frequencies of 420x350 nm sample Excitation amplitude of Hy = 25 Oe Low-resistance High-resistance
Frequency (GHz)
Resonance frequency (GHz)
a
Resistance response of 420x350nm sample during a sequence of DC-response of 420x350 nm sample during 100 ns excitations 100ns c excitations with 58G amplitude and frequencies 3.15GHz, 3.65GHz
80 80
FIG. 5: (Color online) a, Measured resonance frequencies of a 420x350 nm spin-flop bilayer. The intensity map (red and blue for the two ground states) is normalized to the maximum oscillation amplitude for the given EA-field. b, Switching probability for the two ground states subject to 300 ms long excitation pulses with -20 Oe static EA-field applied. Black color denotes the frequency-amplitude points, for which switching in both directions occurred. The inset shows the section of the analytical switching map (Fig. 3) covered in the measurement. c, Real time resistance traces of a sample excited by short microwave pulses of alternating frequencies in resonance with the split optical mode, showing controlled resonant state selection. d, Resonant state selection using resonant microwave pumping of fixed frequency and alternating external EA-field, tuning the two ground states in and out of the optical spin resonance.
citation of fixed amplitude (typically 25 Oe). Figure 5a shows the measured resonance frequencies for the two AP ground states of a SFi with M0 < 0, H1m ≈ 7 Oe, H2m ≈ 3 Oe and t1 −t2 ≈ 0.9 nm (values deduced from quasistatic EA-hysteresis as shown in fig 4c). The splitting of the optical spin resonance at ≈ 4 GHz is in excellent agreement with the predicted behavior (Fig. 3). Figure 5a shows the measured oscillation amplitude as a function of the excitation frequency and applied bias field. The amplitude is normalized with respect to the maximum value for each bias field. This diagram clearly shows that both field-controlled and frequency-controlled switching is possible. Frequency-amplitude-switching maps were measured for both AP states (low- to high-resistance and highto low-resistance) for different excitation pulse durations and static EA-fields. Figure 5b shows the switching maps for both AP states of the soft bilayer under 300 ms pulses of resonant excitation, with Hx = −20 Oe. Switching at frequencies lower than 3.9 GHz only occurs from the low- to high-resistance AP state (fL→H ), while switching at above 3.9 GHz only occurs from the high- to lowresistance state (fH→L ). Shortening the duration of the
excitation pulse requires a higher amplitude for switching the junction, at the same time leading to an increase in the error-rate likely due to a more non-linear and more non-uniform process for higher-amplitude pumping. Nevertheless, even for short excitations, the optical resonance is clearly split and amplitude-frequency regions exist where the switching is strictly one-directional. The weaker higher-order oscillations in the probability maps of Fig. 5b are due to secondary spin-wave excitations in our nanoparticles that are somewhat bigger than the true single-domain limit, which we also observe in our micromagnetic simulations as a small superposition on the well-defined macrospin behavior. Figure 5c shows a realtime trace of the junction resistance during the application of a sequence of microwave pulses of 100 ns duration, of frequency alternating between the split optical spin-resonance peaks. A pair of pulses is sequentially applied at each frequency, demonstrating the strictly unidirectional character of resonant switching at a given frequency (for a given AP state of the SFi). Figure 5d shows a realtime resistance trace (blue) under continuous microwave excitation of fixed frequency
8 near the centre of the optical resonance, with the EAfield of amplitude sufficient for tuning the two AP ground states in and out of the resonance. Between the EApulses the excitation is turned off shortly to confirm the state the sample is in (microwave amplitude shown in red; EA-field in green). The EA-field pulses were generated with an external electromagnet, so their shortest duration was 300 ms. The data show that the field-controlled resonance state selection is highly reliable. Both measurements were repeated many times with the resulting error-rate smaller than 10−3 (1000 measurements were performed without any error). The effect is thus essentially deterministic for the sample and frequency-field parameters used, and still is expected to improve (e.g., in terms of speed) on scaling down the sample size. V.
CONCLUSIONS
We show how external microwave and static biasing fields can be used to tune the energetics and dynamics of a spin-flop system between symmetric and asymmetric be-
∗
Corresponding author:
[email protected] M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau, F. Petroff, P. Etienne, G. Creuzet, A. Friederich, and J. Chazelas. Phys. Rev. Lett., 61, 2472, (1988). 2 G. Binasch, P. Gr¨ unberg, F. Saurenbach, and W. Zinn. Phys. Rev. B, 39, 4828, (1989). 3 M. Julliere. Phys. Lett. A, 54, 225, (1975). 4 J. S. Moodera, Lisa R. Kinder, Terrilyn M. Wong, and R. Meservey. Phys. Rev. Lett., 74, 3273, (1995). 5 B. Dieny, V. S. Speriosu, S. S. P. Parkin, B. A. Gurney, D. R. Wilhoit, and D. Mauri. Phys. Rev. B, 43, 12970, (1991). 6 Th. Gerrits, H. A. M. van den Berg, J. Hohlfeld, L. Bar, and Th. Rasing. Nature, 418, 509, (2002). 7 Anders Bergman, Bj¨ orn Skubic, Johan Hellsvik, Lars Nordstr¨ om, Anna Delin, and Olle Eriksson. Phys. Rev. B, 83, 224429, (2011). 8 C. H. Back, R. Allenspach, W. Weber, S. S. P. Parkin, D. Weller, E. L. Garwin, and H. C. Siegmann. Science, 285, 864, (1999). 9 S. Kaka and S. E. Russek. Appl. Phys. Lett., 80, 2958, (2002. 10 J. C. Slonczewski. J. Magn. Magn. Mater., 159, L1, (1996). 11 M. C. Wu, A. Aziz, D. Morecroft, M. G. Blamire, M. C. Hickey, M. Ali, G. Burnell, and B. J. Hickey. Appl. Phys. Lett., 92, 142501, (2008). 12 E. B. Myers, D. C. Ralph, J. A. Katine, R. N. Louie, and R. A. Buhrman. Science, 285, 867, (1999). 13 A. Brataas, A. D. Kent, and H. Ohno. Nature Mater., 11, 372, (2012). 14 B.N. Engel, J. Akerman, B. Butcher, R.W. Dave, M. DeHerrera, M. Durlam, G. Grynkewich, J. Janesky, S.V. Pietambaram, N.D. Rizzo, J.M. Slaughter, K. Smith, J.J. 1
havior. We demonstrate controlled state selection in a synthetic ferrimagnet using a frequency modulated resonant microwave field or a uniform external biasing field. This effect is explained analytically using a small-signal analysis of the SFi spin dynamics, predicting a tunable splitting of the optical spin resonance, which is in good agreement with our numerical arbitrary signal-strength simulations. The effect is fast, robust, and offers a new way of magnetic switching of spin-flop nanodevices used in such large scale applications as magnetic random access memory. Our analysis shows that the switching process is fast, ∼ 1/2fa ∼ 500 ps, and proceeds through an in-phase (acoustical) spin rotation with superposed out-of-phase (optical) spin oscillations. Experimentally we demonstrate essentially error-free controlled switching down to 100 ns in pulse duration of the resonant microwave excitation. Smaller, more single-domain-like samples (sub100 nm range), with optimized magnetic asymmetry and increased aspect ratio are expected to show significantly faster resonant switching due to reduced generation of unwanted spin-wave modes and higher acoustical frequencies.
Sun, and S. Tehrani. IEEE Trans. Magn., 41, 132, (2005). H. T. Nembach, C. Bayer, H. Schultheiss, M. C. Weber, P. Martin Pimentel, P. A. Beck, B. Leven, and B. Hillebrands. Appl. Phys. Lett., 87, 142503, (2005). 16 D. C. Worledge. Appl. Phys. Lett., 84, 4559, (2004). 17 A. V. Kimel, B. A. Ivanov, R. V. Pisarev, P. A. Usachev, A. Kirilyuk, and Th. Rasing. Nature Phys., 5, 727, (2009). 18 Y.-T. Cui, J. C. Sankey, C. Wang, K. V. Thadani, Z.-P. Li, R. A. Buhrman, and D. C. Ralph. Phys. Rev. B, 77, 214440, (2008). 19 T. Moriyama, R. Cao, John Q. Xiao, J. Lu, X. R. Wang, Q. Wen, and H. W. Zhang. Appl. Phys. Lett., 90, 152503, (2007). 20 S.S. Cherepov, V. Korenivski, and D.C. Worledge. IEEE Trans. Magn., 46, 2112, (2010). 21 V. Korenivski and D. C. Worledge. Appl. Phys. Lett., 86, 252506, (2005). 22 A. Konovalenko, E. Lindgren, S. S. Cherepov, V. Korenivski, and D. C. Worledge. Phys. Rev. B, 80, 144425, (2009). 23 S. S. Cherepov, B. C. Koop, Yu. I. Dzhezherya, D. C. Worledge, and V. Korenivski. Phys. Rev. Lett., 107, 077202, (2011). 24 D. W. Abraham and D. C. Worledge. Appl. Phys. Lett., 88, 262505, (2006). 25 D. C. Worledge. Appl. Phys. Lett., 84, 2847, (2004). 26 S.-Y. Wang and H. Fujiwara. J. Appl. Phys., 98, 024510, (2005). 27 A. M. Kosevich, B. A. Ivanov, and A. S. Kovalev. Physics Reports, 194, 117, (1990). 28 W. J. Gallagher and S. S. P. Parkin. IBM J. Res. Dev., 50, 5, (2006). 29 D. W. Abraham, P. L. Trouilloud, and D. C. Worledge. IBM J. Res. Dev., 50, 55, (2006). 15
9 30
M. C. Gaidis, E. J. O’Sullivan, J. J. Nowak, Y. Lu, S. Kanakasabapathy, P. L. Trouilloud, D. C. Worledge, S. Assefa, K. R. Milkove, G. P. Wright, and W. J. Gal-
lagher. IBM J. Res. Dev., 50, 41, (2006).
10
Supplementary: Controlled resonant switching of a synthetic ferrimagnet
Macrospin theory
Considering the free layers of the synthetic ferrimagnet (SFi) as single-domain particles, with the the z-axis outof-plane, the EA and a uniaxial anisotropy field along the x-axis, the total magnetic energy (E) of the two free layers can be written as1 : E = 2πMS2
2 X
nature. Since in a thin magnetic film the in-plane demagnetising factors are proportional to their thickness, they can be rewritten as Ni{x,y} = 1 − (−1)i ε N{x,y} , Niz = 1 − Nix − Niy , where
Vi Nix m2ix + (Niy + hu )m2iy + Niz m2iz
N{x,y} =
i,j=1;i6=j
+ γjx m1x m2x + γjy m1y m2y + γjz m1z m2z −2(hx + hm ix )mix − 2hy cos (ωt)miy }, where MS is the saturation magnetisation (assumed same in both layers); Vi = πabti /4 the volume of layer i with a, b and ti the length, width and thickness of that layer, respectively; Niα the demagnetising P factors of layer i in the α-dimension with α Niα = 1 and Ni{x,y} = n{x,y} ti /b where n{x,y} are the reduced factors2 ; mi the cartesian unit magnetisation vector of the i-th layer; hu = Hu /4πMS the normalised intrinsic uniaxial anisotropy field along the easy-axis; hy = Hy /4πMS the normalised amplitude of the applied ACfield in the y-direction; ω the frequency of the AC-field; hx = Hx /4πMS the normalised external easy-axis DCm field; hm i = Hi /4πMS the normalised fringing field acting on layer i in the x-direction; γiα = rα Niα the interlayer exchange field from layer i acting on the other layer in the α-direction3 . In our case γiα is solely of dipolar
(t1 + t2 ) n{x,y} t1 − t2 , ε= . 2b t1 + t2
The same can be done for the volume and interlayer exchange field of the individual layers: Vi = 1 − (−1)i ε V /2; γi{x,y} = 1 − (−1)i ε γ{x,y} , For the description of the dynamics of the magnetisation of the system, it is useful to convert the magnetisation into polar coordinates: mi = ( cos ϕi sin θi ,
sin ϕi sin θi ,
cos θi ),
Since in the used geometry the thicknesses are much smaller than the lateral dimensions and the sample is longer along the x-dimension than the y-dimension, the following holds: Nix < Niy Niz ≈ 1;
θi = π/2 + ξi , |ξi | 1,
Hence, cos(θi ) ≈ −ξi . The terms rz , ξ1 , ξ2 are minimal and therefore γz ξi ξj can be neglected:
hu W − W0 hu 2 = − (1 + ε ) Ny − Nx + cos 2Φ cos 2χ + 2ε Ny − Nx + sin 2Φ sin 2χ πMS2 1 + ε2 2 (S1) − (1 − ε2 ) [(γy − γx ) cos 2Φ + (γy + γx ) cos 2Φ] − 4(hx + hf + εhd ) cos Φ cos χ 1 + 4[ε(hx + hf ) + hd ] sin Φ sin χ − 4hy cos ωt(sin Φ cos χ + ε cos Φ sin χ] + (m2z + lz2 + 2εmz lz ), 2
where W0 = π[(1 + ε2 )(Ny + Nx ) + hu ]MS2 ; Φ, χ are the normal coordinates and are given by Φ = (ϕ1 + ϕ2 )/2, χ = (ϕ1 − ϕ2 )/2; mz = ξ1 + ξ2 , lz = ξ1 − ξ2 , here mz and lz are the normalised z-components of the magnetisation vectors of the resulting pair of layers. Further m the following relations were used: hf = (hm 1 + h2 )/2, m hd = (hm − h )/2. 1 2 The Lagrange density is then given by4 : 2
L=
1X¯ hMS ∂ϕi (1 − cos θi ) (1 − (−1)i ε) − W. 2 i=1 2µB ∂t
Hence, L dΦ dχ = (1 + mz ) +ε 2πMS2 dτ dτ dχ dΦ + lz +ε − W (Φ, χ, mz , lz ), dτ dτ where τ = t · ω0 , ω0 = 8πMS µB /¯h, Ω = ω/ω0 . Therefore, the dynamics of the system are described by:
11
d2 (Φ + εχ) = hy cos Ωτ (cos Φ cos χ − ε sin Φ sin χ) − (hx + hf + εhd ) sin Φ cos χ − [ε(hx + hf ) + hd ] cos Φ sin χ dτ 2 o 1 n hu 2 2 − (1 + ε ) (Ny − Nx ) + hu cos 2χ + (1 − ε )(γy − γx ) sin 2Φ − ε Ny − Nx + cos 2Φ sin 2χ, 2 2 d2 (S2) (χ + εΦ) = hy cos Ωτ (ε cos Φ cos χ − sin Φ sin χ) − (hx + hf + εhd ) cos Φ sin χ − [ε(hx + hf ) + hd ] sin Φ cos χ dτ 2 o 1 n hu − (1 + ε2 ) (Ny − Nx ) + hu cos 2Φ − (1 − ε2 )(γy + γx ) sin 2χ − ε Ny − Nx + sin 2Φ cos 2χ, 2 2
mz = 2
dΦ ; dτ
From these equations it is also easy to determine that the ground states in the absence of the oscillating yfield (hy = 0) is given by Φ = π/2, χ = ±π/2 which corresponds to the antiferromagnetic vector values l = (±2, 0, 0). A.
Resonance frequency
To switch the SFi resonantly between its two ground states, it is necessary to determine the resonancefrequencies of the ground states. If the amplitude of the oscillating field is small (|hy | 1), then the magnetic moments of the SFi deviate only minimal from their ground state: π π Φ = + Φ1 ; χ = χ0 + χ1 ; χ0 = ± ; |Φ1 |, |χ1 | 1 2 2 (S3) Substituting the relations of (S3) into Eqs. (S2) gives: d2 Φ1 d2 χ1 2 + Ω Φ + ε + Ω212 χ1 = εhy cos Ωτ sin χ0 , 1 11 dτ 2 dτ 2 d2 χ1 d2 Φ1 2 + Ω χ + ε + Ω221 Φ1 = hy cos Ωτ sin χ0 , 1 22 dτ 2 dτ 2 (S4) Ω2ii = (1 + ε2 )(Ny − Nx ) + (1 − ε2 )(γx + (−1)i γy ) + [ε(hx + hf ) + hd ] sin χ0 , Ω2ij = 2ε(Ny − Nx + hu /2) − (hx + hf + εhd ) sin χ0 , where i 6= j. The eigenfrequencies or resonancefrequencies of a system of equations as in Eqs. (S4) is easy to solve, and is given by n 2 Ω± = Ny −Nx +γx +hu −hd sinχ0 ± (1−ε2 )γy2 o1/2 2 + [(hx + hf ) sin χ0 − ε(Ny − Nx − γx )] where ”−” stands for acoustical (in-phase) and ”+” for optical (out-of-phase) resonance frequencies. Hence, the
lz = 2
dχ . dτ
criterion for Hx for degenerate resonance in both states is given by s hx = −hf − hd
1−
h2d
(1 − ε2 )γy2 , − − Nx − γx )2 ε2 (Ny
where all the terms are functions of (t1 , t2 ) or (H1m , H2m ). The difference in optical frequency of the two ground states (∆foptical = ∆Ω+ · ω0 /2π) can be approximated by the linearisation of the frequencies around ε = 0 and hd = 0: (Ny −Nx −γx ) γy2 +(hx +hf )2
hd ± ε(hx +hf ) √
∆Ω± ≈ r q Ny −Nx +γx +hu ± γy2 + (hx +hf )2
B.
Resonant stability
To determine the switching mechanism for the SFi using resonant excitation we analyse the behaviour and stability of the system for large excitation amplitudes. For this we only consider a fringing field asymmetry (which has the largest contribution to the splitting of resonance) and take the thicknesses of the two free layers to be equal (t1 = t2 ⇒ ε = 0). To simplify the derivation, the external DC field is set to hx = −hf (This is the external field at which the splitting is maximal). When an external AC-field of suitably high frequency is applied along the hard-axis (y-axis) of the elliptical magnetic particles, the magnetic moments begin to oscillate in the optical mode, which results in a modulation of χ. At the same time, Φ only deviates slightly from its ground state Φ0 = π/2. Therefore, it is a good approximation to take ϕ = Φ − π/2 (|ϕ| 1). Taking all the above assumptions in consideration, Eqs. (S2) can be rewritten as
12
d2 ϕ 2 + Ωa − 2(Ny − Nx + hu ) cos2 χ dτ 2 − hd sinχ + hy cos Ωτ cosχ ϕ = 0,
Substituting Eq. (S10) into Eq. (S9) results in a fourth order differential equation: (S5)
d2 χ − Ω2o sinχ cosχ + hd cosχ + hy cos Ωτ sinχ = 0, (S6) dτ 2 with Ωa and Ωo the characteristic frequency of small amplitude ’acoustical’ and ’optical’ oscillations, respectively, in p a symmetric system and are given by Ωa = (Ny − Nx + hu ) − (γy − γx ) and Ωo = p (Ny − Nx + hu ) + (γy + γx ). Provided that Ωa Ωo , (which is valid for elliptical particles with aspect ratio of approximately 1) the variation in ϕ can be regarded as ’slow’ compared to the ’fast’ oscillations of χ. If the frequency of the applied external field is near the optical resonance frequency Ωo , then averaging the χ- and Ωterms in Eq. (S5) over one period is justified and yields 2 d2 ϕ + Ωeff · ϕ = 0, a 2 dτ 2 Ωeff a
=
Ω2a
− 2 (Ny − Nx + hu ) ·
f 4 1. This requirement is not too restrictive and allows to consider nonlinear oscillations with high amplitude. Following the standard procedure for finding the resonance in a nonlinear system, such as Eq. (S11), the solution is represented in the form f = A cos (Ωτ + δ) + ζ,
(S7)
cos2
χ
− hd · sin χ + hy · cos Ωτ cos χ,
(S8)
where the bar stands for averaging over one period of the ’fast’ oscillation. Using Eq. (S7) the stability of different magnetic configurations resulting from the action of the AC-field excitation can be determined. Whenever Ωeff a becomes complex, the state of the particle becomes unstable, which can result in switching. To evaluate Eq. (S6) for external excitations at the optical frequency we add a phenomenological term describing the dissipative processes in the system: d2 χ dχ + 2λΩo − Ω2o sin χ cos χ dτ 2 dτ +hd cos χ + hy cos Ωτ sin χ = 0.
(S9)
Here λΩo is the parameter characterising the dissipation of energy in the system. Its value can be associated with a dissipative Gilbert parameter. Starting in the ground states (Φ = π/2 and χ = ±π/2) and turning on the AC-field results in periodic oscillations of χ, which will be described using a new variable f , connected with χ by the following relation: χ = χ0 + 2 arctan(f ),
d2 d 2 1 + f2 + 2λΩ + Ω − h sin χ o d 0 f o dτ 2 dτ # " 2 hy df 2 2 + Ωo f f + −2 sin χ0 cos Ωτ 1−f 4 = 0. dτ 2 (S11) To reduce the degree of nonlinear terms in Eq. (S11), the accuracy of the theory is limited to
χ0 = ±π/2
(S10)
Here and further on the upper sign corresponds to the state in which the magnetic moments fluctuate in the vicinity of ~l = (2, 0, 0), and the lower sign to fluctuations in the vicinity of ~l = (−2, 0, 0).
(S12)
with A the amplitude of the main harmonic, δ the phaseshift with respect to the applied field, and ζ an additional term taking into account anharmonic corrections. When the external excitation frequency Ω is close to the resonance frequency, then the main harmonic oscillations dominate and |ζ| A, in which case the anharmonic corrections can be neglected. Substituting Eq. (S12) into Eq. (S11) and only considering the terms proportional to the main harmonics, (sin (Ωτ ) and cos (Ωτ )), the resulting amplitude and phase-shift of the oscillations become:
2 h2y 5A2 3A2 2 2 ≈ Ω 1+ − Ωo −hd sinχ0 1− 4A2 4 4 2 2 A 2 + (2λΩΩo ) 1 + , 4 (S13) 2λΩΩo (1 + A2 /4) . Ω2 (1+5A2/4)−(Ω2o −hd sinχ0 )(1−3A2/4) (S14) From these equations it follows that at the excitation frep quency of Ω = (Ω2o −hd sin χ0 )(1−3A2 /4)/(1+5A2 /4) the system is in resonance when the dissipation is weak. In this case the amplitude reaches a maximum Amax defined by Amax (1 + A2max /4) = hy /4λΩΩo . Substituting Eq. (S12) into (S10) into (S8) and taking the appropriate averages gives the effective ’acoustic’ resonance frequency and with that the stability of the ground states: tan δ =
13
Ωeff a
2
= Ω2a −
(Ny − Nx + hu )4A2 − hd sin χ0 (1 + A2 )3/2
√
When the oscillation amplitude A substantially increases, the right hand side of Eq. (S15) becomes negative, which makes the respective state unstable. The difference in the resonance frequency for the two ground states ~l = (2, 0, 0) and ~l = (−2, 0, 0) is proportional to the asymmetry in the fringing field (hd ). Therefore, it is possible to determine the region in the frequency-amplitude phase space, in which one state is unstable while the other is stable. It is this asymmetry-induced selectivity in, for example, frequency for a given amplitude that allows to uniquely select a given AP state of the SFi. To determine the frequency-amplitude range where only one of the states is stable, the critical amplitude at which Ωeff a becomes imaginary needs to be determined for the the two AP states. To do this, Eq. (S15) needs to be solved for the case where Ωeff a = 0. Assuming that the fringing field difference and the AC-field amplitude are small, then the last two terms in (S15) can be neglected, and the critical amplitude Ac becomes: 4A2c 1 + A2c
−3/2
=
(Ny − Nx + hu ) − (γy − γx ) , (Ny − Nx + hu )
which reduces to a cubic equation and is straightforward to solve. For our samples the equation has three real solutions, of which one, A2c = 0.118, satisfies the criterion of applicability, A4c 1. Once Ac is determined, Eqs. (S13) and (S14) can be used to determine the frequency-amplitude values for which each state changes between stable and unstable. Hence, a ground state is only stable if s r 2 hy 3A2c hd sinχ0 A2 2 1− c , 1− γ + ≤ A +λ f c 4Ω2o 4 2Ω2o 4 where γf = (Ω − Ωc )/Ωc with the critical frequency Ωc = p 2 Ωo (1 − 3A2c /4)/(1 + 5A2c /4). The above result takes into account that |γf | 1. C.
Switch and spin-flop fields
The switching and spin-flop fields that are determined by measuring the EA hysteresis of the sample are used to calculate the approximate thicknesses and fringing-fields. The derivation of this formula is quite straightforward. The switching and spin-flop occur when the initial ground state is no longer an energy-minimum. Starting from the energy-density given by Eq. (S1), the switching and spinflop fields can be determined by solving hx for 2 2 ∂2W ∂2W ∂ W − = 0. (S16) ∂Φ2 ∂χ2 ∂Φ∂χ
2 hy 1 − 1 − 2 cos δ sin χ0 . 1− √ A 1 + A2 1 + A2
(S15)
Here the out-of-plane terms are neglected since we are only interested in the in plane AP ground states in which case Eq. (S16) is the resulting equation. The two ground states are given by Φ = Φ0 = π/2 and χ = χ0 = ±π/2. In this case the solutions of Eq. (S16) are: hx = ε(Ny − Nx − γx ) sin χ0 − hf q ± (Ny − Nx + γx + hu − hd sin χ0 )2 − (1 − ε2 )γy2 . (S17) Now since the dipole coupling is strong and the aspect ratio relatively close to 1, Ny − Nx − γx < 0. Therefore the sign of ε determines which of the states have positive or negative spin-flop or switch fields. Hence, in order to accurately give the relation for the different transition fields (H1± , H2± ), equation (S17) has to be slightly altered: Hi± = ∓(−1)i |ε|(Ny −Nx −γx ) − hf 4πMS s 2 ε i ± Ny −Nx +γx +hu ±(−1) hd −(1 − ε2 )γy2 . |ε| (S18) Now to be able to determine the individual thicknesses and fringing fields, Eq. (S18) has to be rewritten in the fundamental parameters, such that none are dependent on thicknesses or fringing fields. This results in the following set of equations: H m +H2m |t1 −t2 | Hi± (ny −(1+rx )nx )− 1 = ∓(−1)i 4πMS 2b 8πMS ( Hu (ny −(1−rx )nx ) + ± (t1 +t2 ) 2b 4πMS ) r n 2 1/2 m m 2 y y i t1 − t2 H1 − H2 ±(−1) −t1 t2 , |t1 − t2 | 8πMS b
|t1 − t2 | =
H1+ − H1− − H2+ + H2− b , (ny − (1 + rx )nx ) 8πMS
H1m + H2m = −(H1+ + H1− + H2+ + H2− )/2.
14
∗ 1 2 3
Corresponding author:
[email protected] D. C. Worledge Appl. Phys. Lett. 84, 4559 (2004). D. C. Worledge Appl. Phys. Lett. 84, 2847 (2004). S.-Y. Wang & H. Fujiwara J. Appl. Phys. 98, 024510 (2005).
4
A. M. Kosevich, B. A. Ivanov & A. S. Kovalev Physics Reports 194, 117 (1990).
